https://file.aiquickdraw.com/mathbot/file/1726751799199zmcvg53p.jpg

https://file.aiquickdraw.com/mathbot/file/1726751799199ak3k47tl.jpg

The function given is:

\[
f(x) = -\frac{5}{16}(x - 4)^2 + 5
\]

This is a quadratic function in vertex form, which provides key information about the graph:

1. **Vertex**: The vertex form \( f(x) = a(x - h)^2 + k \) shows the vertex of the parabola directly. Here, \( h = 4 \) and \( k = 5 \), so the vertex is at \( (4, 5) \).

2. **Direction of Opening**: The coefficient \( a = -\frac{5}{16} \) is negative, which means the parabola opens **downward**.

3. **Stretch/Compression**: Since \( |a| = \frac{5}{16} \) is a fraction, the graph is vertically compressed compared to a standard parabola.

### Matching to the Graphs

Looking at the images you provided:

- The first graph shows a parabola opening **upward**, so it does not match.
- The second graph shows a parabola opening **downward**, and it has a maximum point around \( (4, 5) \), which matches the vertex. This is the correct graph.

Thus, the second graph in the second image corresponds to the function \( f(x) = -\frac{5}{16}(x - 4)^2 + 5 \).

Would you like further details or have any questions?

---

Here are 5 related questions for further practice:

1. How would the graph change if the coefficient were positive, like \( \frac{5}{16}(x - 4)^2 + 5 \)?
2. What happens to the graph if the vertex shifts from \( (4, 5) \) to \( (-2, 3) \)?
3. How does the width of the parabola change as the coefficient \( a \) increases or decreases in magnitude?
4. How do you find the axis of symmetry for a quadratic function in vertex form?
5. What effect does changing the constant \( +5 \) in the equation have on the graph?

**Tip**: Always focus on the sign and magnitude of the leading coefficient to determine the direction and width of a parabola.

Match the function f(x) = -5/16(x - 4)^2 + 5 to the graph it represents.

Understand how to match the quadratic function f(x) = -5/16(x - 4)^2 + 5 to its corresponding graph. This problem covers vertex form, direction of parabolas, and key graph characteristics like vertex and axis of symmetry. Suitable for high school students in Grades 9-11.

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